Questions tagged [numerics]
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732
questions
1
vote
1answer
37 views
Type of Rosenbrock method by its coefficients
A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients:
...
2
votes
0answers
41 views
evaluating $\coth(x) - 1/x$ for real x, on 2 “pieces”
The function $\coth(x) - 1/x$ has a removable singularity at 0.
Its Taylor series is:
$$
\coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots
$$
I would like to evaluate the ...
2
votes
1answer
48 views
pdepe or Crank-Nicolson? How much is pdepe good?
I am beginner in MATLAB and similar. I sow and discussed with my professors doing simulations some times: they wrote down a lot of calculus, most of them using Crank-Nicolson Method and so implement ...
0
votes
0answers
52 views
Comparison of diffusion time - theoretical value vs computed
This is a follow up to my previous post
I've been trying to compare the diffusion time obtained from theoretical
derivation(answered in my previous post) and what is obtained computationally, for a ...
2
votes
1answer
100 views
mesh dependence of numerical adjoint solution
I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are ...
0
votes
2answers
167 views
What are the most important theorems in computational science? [closed]
I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this:
"The Lax-Milgram Lemma is one ...
2
votes
0answers
33 views
How to account for a corner node with zero-flux condition at an extrapolated distance
I am trying to implement a numerical solver and am having troubles dealing with boundary conditions, especially in the corners.
I have a 2D mesh, and on the left I have a Dirichlet condition, on the ...
14
votes
1answer
953 views
Conserving Energy in Physics Simulation with imperfect Numerical Solver
I am creating a C++ Physics Simulation where I need to move an rigid body through an acting force field.
Problem: simulation does not conserve energy.
Quesiton: abstractly, how is conservation of ...
2
votes
0answers
27 views
Computing convolution of two characteristic function over a 1D Cartesian mesh
I am trying to compute the convolution of two characteristic functions over a Cartesian mesh. First, I define my Cartesian mesh of the interval $[0,1]$ as follows
$$
x_{i} = i \Delta x, i = 0, 1, 2\...
0
votes
0answers
137 views
Ising model simulation offset critical temperature and interal ernergy
I'm writing a code for the Ising model using WHAM (the weighted histogram analysis method)，But it seems to produce critical temperature and internal energy wrong.
(newest rewritten code is below)
<...
1
vote
0answers
53 views
Stably solve transport equation with source term
I am trying to solve a transport equation of the form for the variable $\psi(t,r)$
\begin{equation}
\partial_t\psi-\alpha(r)\partial_r\psi-\beta(r)^2\psi-f(t,r)=0
,
\end{equation}
where I am solving ...
0
votes
1answer
82 views
Solution of thermal analysis using finite element
I want to solve a thermal analysis using finite elements. The governing equation is $$C \frac{dT}{dt}+K T = Q$$.
When using backward differencing for time, the resulting equation is quite straight ...
4
votes
1answer
81 views
Modelling flow through pipe networks
I'm trying to educate myself on modelling solute flows through pipe networks.
This is a follow up of my previous post here
$$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$
While ...
2
votes
1answer
82 views
Does mass balance hold in convective diffusion
I'm trying to understand how convection-diffusion equations are solved in pipe flow modules available in CFD solvers.
$$
\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \...
0
votes
1answer
67 views
Solving differential equation in Python with discretized variable coefficients
I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie.
In this case the Runge-Kutta step size is fixed by the frequency in the time ...
3
votes
0answers
65 views
Numerical integration with singularity term
In https://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities, the author explains the subtraction method to get rid of singularities when performing numerical integration.
The ...
6
votes
2answers
1k views
Runge-Kutta in the presence of an attractor
Suppose you are solving a system of equations numerically that possesses an attractor (no matter the initial conditions set, all the different solutions will approach a specific set of values that ...
6
votes
0answers
97 views
How to check if my stiffness matrix is correct
I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
5
votes
2answers
123 views
Is the diffusion equation with Neumann and Dirichlet BCs well-posed?
I am considering the following diffusion equation:
$$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$
over a grid ...
2
votes
1answer
43 views
Numerical integration of the dataset of a function
The energy equation for a spherically symmetric system is given by
$$\mathscr{E}=\frac{v^2(r)}{2}+\frac{c_s^2(r)}{\gamma-1}+\phi(r)$$
where $\mathscr{E}$ is the total energy, $v$ is the velocity of ...
2
votes
1answer
67 views
Defining Current Density in a FEM model (MATLAB)
I'm attempting to solve the Poisson equation in 3D for a magnetic vector potential in the presence of a current source. To validate my code, I'm initially looking to reproduce the model described in ...
-1
votes
1answer
39 views
Integrate using Composite Simpson's rule
In a question, we have been given the speed of a car at time t= 0,2,4,6,.......,20 minutes.But it asks us to approximate the distance travelled by the car in 30 minutes using Composite
Simpson's rule. ...
1
vote
1answer
90 views
Solve linear system with Newton-Raphson method
Is it possible to solve a linear matrix system $A x = b$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?
-1
votes
1answer
54 views
Recursive Algorithm to Calculate Determinant via Expansion of Minors in C#
I have been recently trying to attempt to write an algorithm in C# that would calculate the determinant of a matrix via recursion using the expansion of minors method. I understand that there are ...
2
votes
0answers
94 views
What exactly is the cause(s) of blow-up for too-large step size in a method like RK4?
I have been working on creating a few home-made numerical methods, and I am using them to visualize text-book problems from my Strogatz dynamics textbook. It feels like a good way to learn numerical ...
0
votes
0answers
22 views
Interpreting results of using no-flux boundary condition
I am solving for solute transport in 1 D.
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$
No-flux boundary condition is imposed at both the ...
1
vote
1answer
69 views
Implementing Robin Boundary condition (finite difference)
I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D.
In the following system,
$$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
2
votes
0answers
85 views
Extracting FEM matrices in matlab pde toolbox
I am trying to follow the dynamic linear elasticity in Matlab, link here. My goal is to extract the FE Matrices using the function assembleFEMatrices in matlab and solve the resulting system of second-...
-1
votes
1answer
109 views
Imposing periodic boundary condition for linear advection equation - Node problem
I've spent the whole day trying to figure out what is the correct way to impose (and implement) periodic boundary conditions $u(0,t)=u(1,t)$ for all $t>0$ for the simple advection equation $u_t + v ...
0
votes
1answer
83 views
Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation
I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this
$$
\frac{\partial U}{\partial t} + ...
1
vote
1answer
118 views
Runge-Kutta fourth order method. Integrating backwards
I am using a Runge-Kutta fourth order method to solve numerically the usual equation of motion of a background scalar field in curved spacetime with a quartic potential:
$\phi^{''}=-3\left(1+\frac{H^{...
1
vote
1answer
73 views
How to Break Coupled ODEs down to first order for Runge-Kutta
My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
2
votes
1answer
62 views
How to robustly and numerically expand a $k$-order polynomial in two variables defined on a polygon domain?
Given a $k$-order polynomial in two variable $p(x, y)$ defined on a polygon domain $K$. And I want to numerically expand it to the following form
$$
p(x, y) = c_0 + c_1 x + c_2 y + c_3 x^2 + c_4 xy + ...
1
vote
0answers
51 views
Numerical Method for Equation System of two depending Equation Systems
I am searching a solution method for the following equation system of equation systems:
Let $A \in \mathbb{R}^{n \times n}$ be an invertible Matrix, $f, b_1, b_2 \in\mathbb{R}^n$ given vectors and $ ...
6
votes
1answer
117 views
Computing square root of diag(u)-uu'?
I need an efficient way to take square root of a matrix which is a sum of diagonal matrix and rank-1 matrix.
More specifically it's the following matrix
$$A=D-uu'=\text{diag}(u)-uu'$$
Where entries ...
1
vote
2answers
48 views
Numerical integral with symbolic integral in exponent
Many times in fourier approximation we come across integrals such as
$$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
3
votes
1answer
53 views
Bounding error of float32 matrix multiplication
Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements.
Is there a way to estimate/bound this error without resorting to higher ...
2
votes
1answer
47 views
Passing data as arguments in ODE45
I need to import data from file in order to describe the structure of a network.
I used the following:
weights = readtable('weights192.txt');
W = weights{:,:};
...
2
votes
0answers
56 views
Numerical integration of SDE: choice of $dt$ and algorithm
I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context:
$$dX_{t} = a X_{t} dt + b X_{t} dW$$
where $X_{t}$ is my stochastic varible, $dt$ is my ...
1
vote
1answer
43 views
Weighted moving variance
i have a time-series and, in analogy with exponentially weighted moving average, i would like to compute the exponentially weighted moving standard deviation or variance in an efficient, numerically ...
2
votes
0answers
39 views
Inverses of many standard subspaces of one large matrix
i have a large rectangular invertible matrix M (about 5000x5000) and i have a loop in which i do the following for each iteration i (there are about 6000 iterations):
i am given a subspace S_i (which ...
1
vote
0answers
72 views
Runge-Kutta for PID and system in separate calculations without filter
I need to calculate a closed-loop system in Python; specifically, obtain the PID response and then use the output to obtain the system response sample-by-sample with my own loop.
For this, I am ...
0
votes
0answers
64 views
How to implement the following Finite Element method for Burgers' equation?
I am trying to replicate this result.
It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results....
0
votes
1answer
87 views
Well-posedness of Navier-Stokes equation
Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition??
Can someone ...
7
votes
1answer
122 views
Numerically estimating expected value of f(x) when x is normally distributed
I need to estimate
$$
\mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx
$$
for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the ...
5
votes
0answers
48 views
Evaluate Nth root of a rational to a correctly rounded float
Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible.
I want to evaluate an expression in the ...
1
vote
0answers
88 views
How to compute the following Crank-Nicolson scheme for the viscous Burgers' Equation?
I am attempting to replicate results from this article.
I'm not sure why but my results are completely different and wrong. For example, the exact solution with parameters ($x=0.1$, $T=0.01$, $Re=0....
1
vote
0answers
79 views
What's the more efficient way to solve this matrix equation?
This is intended to be a more generic question not about a specific system. Given a hermitian matrix $H(x_1,\dots,x_n)$ depending non-linearly on some real parameters $x_1,\dots,x_n$. We want these to ...
4
votes
0answers
100 views
Integrators for Nonlinear/Stiff PDE
It was suggested I ask this question in this section. Anyway:
I have a particular nonlinear PDE of the form
$$
u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t))
\tag{1}
$$
Where f is some nonlinear function. With ...
1
vote
0answers
48 views
WENO scheme on curvilinear coordinates
I've been developing a curvilinear FVM code. So far I've implemented the PPM scheme and am looking into adding WENO schemes. So far I've been discretizing the grid metrics using a second-order central....
